The answer is not as obvious as it may seem. On the one hand, we can say that, to the extent that both religions identify the God disclosed in the scriptures of Judaism as their God, then, yes, both religions worship the same deity. Both religions share some of Judaism’s scriptural narratives, though in Islam these narratives tend to be told rather differently than in the Hebrew Bible/Old Testament. Both religions revere Jesus, though in very different ways, and both have concepts of creation, sin, righteousness, and judgment. One could name other similarities as well.
So we worship the same God, though we identify many of the attributes of God differently from one another. Think of the matter this way: Imagine you and I are at a party together, and you ask me, “Which one of these people is the bishop ?” I respond that he is the man wearing a blue suit, standing next to the potted plant, and holding a glass of water. Now you know who the bishop is, and you could approach him, introduce yourself, and ask him all the questions you’ve ever wanted to ask a bishop. The thing is, while he is wearing a blue suit, he’s not standing next to a potted plant. He’s standing next to a fake plant. And he’s not holding a glass of water. He’s holding a glass of Sprite. In this case, I’ve still identified the bishop for you, but some of the descriptors I’ve used to identify him are incorrect.
This is rather like how Christians, Muslims, and Jews understand their differences in relation to the God they worship. Adherents of these religions believe that the others identify some of the attributes and actions of God correctly, but not all of them. So, for example, all three religions hold that God is eternal, transcendent, and self-revelatory. All three would say that God has given humankind moral instruction. But not all three would say that this same God is revealed most perfectly through the Quran or that God became flesh in Jesus Christ. Not all would identify Israel’s place in God’s saving work in the same way.
And this is where things get prickly. For Christians and Muslims, the attributes of God come to bear in very significant ways on their understanding of such issues as sin and judgment, atonement (or the lack thereof), eternal life, and ethical behavior. At the very heart of Christianity are the claims that God, who is Father, Son, and Holy Spirit, became incarnate, died an atoning death on the cross, and then rose from the dead. These same claims are utterly inconsistent with the Islamic understanding of Allah and his relationship to humankind. For adherents of either religion, to gloss over these distinctives is to ignore important matters that give our lives meaning and shape our worldview.
The claim, then, that Christians and Muslims worship the same God is true–but we should not overstate its significance. God is, as they say, in the details.